LMFDB - Embedded newform 1323.4.a.bk.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504 $ x^7 - x^6 - 43x^5 + 10x^4 + 513x^3 + 258x^2 - 936*x - 504
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.76215$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.76215 q^{2} +14.6781 q^{4} -18.7013 q^{5} +31.8020 q^{8} -89.0586 q^{10} +59.4578 q^{11} +12.2112 q^{13} +34.0213 q^{16} -11.5215 q^{17} -49.8054 q^{19} -274.500 q^{20} +283.147 q^{22} -125.792 q^{23} +224.740 q^{25} +58.1516 q^{26} +228.511 q^{29} +189.567 q^{31} -92.4014 q^{32} -54.8669 q^{34} +33.1458 q^{37} -237.181 q^{38} -594.740 q^{40} +524.149 q^{41} +234.349 q^{43} +872.727 q^{44} -599.039 q^{46} +273.742 q^{47} +1070.25 q^{50} +179.237 q^{52} +255.936 q^{53} -1111.94 q^{55} +1088.20 q^{58} -168.833 q^{59} +195.141 q^{61} +902.749 q^{62} -712.200 q^{64} -228.366 q^{65} +515.148 q^{67} -169.113 q^{68} +319.048 q^{71} -635.518 q^{73} +157.845 q^{74} -731.048 q^{76} -852.002 q^{79} -636.244 q^{80} +2496.08 q^{82} +264.419 q^{83} +215.467 q^{85} +1116.01 q^{86} +1890.88 q^{88} +914.197 q^{89} -1846.38 q^{92} +1303.60 q^{94} +931.428 q^{95} +455.582 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 31 q^{4} + q^{5} + 84 q^{8} - 12 q^{10} + 98 q^{11} + 124 q^{13} + 139 q^{16} - 30 q^{17} - 182 q^{19} + 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} + 245 q^{26} + 323 q^{29} - 26 q^{31}+ \cdots + 3730 q^{97}+O(q^{100}) $ 7 * q + q^2 + 31 * q^4 + q^5 + 84 * q^8 - 12 * q^10 + 98 * q^11 + 124 * q^13 + 139 * q^16 - 30 * q^17 - 182 * q^19 + 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 + 245 * q^26 + 323 * q^29 - 26 * q^31 + 398 * q^32 - 114 * q^34 - 112 * q^37 - 1015 * q^38 + 147 * q^40 + 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 - 288 * q^47 + 2576 * q^50 + 1075 * q^52 + 1353 * q^53 - 156 * q^55 - 81 * q^58 - 165 * q^59 - 56 * q^61 + 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 - 2625 * q^68 + 792 * q^71 - 1487 * q^73 + 2736 * q^74 - 1952 * q^76 - 1273 * q^79 + 2501 * q^80 + 2049 * q^82 + 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 - 1058 * q^89 + 3834 * q^92 - 1653 * q^94 + 3260 * q^95 + 3730 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.76215	1.68367	0.841837	−	0.539732i	$-0.181474\pi$
$2$	4.76215	1.68367	0.841837	+	0.539732i	$0.181474\pi$
$3$	0	0
$3$	0	0
$4$	14.6781	1.83476
$5$	−18.7013	−1.67270	−0.836350	−	0.548196i	$-0.815315\pi$
$5$	−18.7013	−1.67270	−0.836350	+	0.548196i	$0.815315\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	31.8020	1.40546
$9$	0	0
$10$	−89.0586	−2.81628
$11$	59.4578	1.62975	0.814873	−	0.579639i	$-0.196807\pi$
$11$	59.4578	1.62975	0.814873	+	0.579639i	$0.196807\pi$
$12$	0	0
$13$	12.2112	0.260521	0.130261	−	0.991480i	$-0.458419\pi$
$13$	12.2112	0.260521	0.130261	+	0.991480i	$0.458419\pi$
$14$	0	0
$15$	0	0
$16$	34.0213	0.531583
$17$	−11.5215	−0.164374	−0.0821872	−	0.996617i	$-0.526191\pi$
$17$	−11.5215	−0.164374	−0.0821872	+	0.996617i	$0.526191\pi$
$18$	0	0
$19$	−49.8054	−0.601376	−0.300688	−	0.953723i	$-0.597216\pi$
$19$	−49.8054	−0.601376	−0.300688	+	0.953723i	$0.597216\pi$
$20$	−274.500	−3.06900
$21$	0	0
$22$	283.147	2.74396
$23$	−125.792	−1.14041	−0.570204	−	0.821503i	$-0.693136\pi$
$23$	−125.792	−1.14041	−0.570204	+	0.821503i	$0.693136\pi$
$24$	0	0
$25$	224.740	1.79792
$26$	58.1516	0.438633
$27$	0	0
$28$	0	0
$29$	228.511	1.46322	0.731610	−	0.681723i	$-0.238769\pi$
$29$	228.511	1.46322	0.731610	+	0.681723i	$0.238769\pi$
$30$	0	0
$31$	189.567	1.09830	0.549150	−	0.835724i	$-0.314951\pi$
$31$	189.567	1.09830	0.549150	+	0.835724i	$0.314951\pi$
$32$	−92.4014	−0.510451
$33$	0	0
$34$	−54.8669	−0.276753
$35$	0	0
$36$	0	0
$37$	33.1458	0.147274	0.0736370	−	0.997285i	$-0.476539\pi$
$37$	33.1458	0.147274	0.0736370	+	0.997285i	$0.476539\pi$
$38$	−237.181	−1.01252
$39$	0	0
$40$	−594.740	−2.35092
$41$	524.149	1.99654	0.998272	−	0.0587584i	$-0.0187141\pi$
$41$	524.149	1.99654	0.998272	+	0.0587584i	$0.0187141\pi$
$42$	0	0
$43$	234.349	0.831115	0.415558	−	0.909567i	$-0.363586\pi$
$43$	234.349	0.831115	0.415558	+	0.909567i	$0.363586\pi$
$44$	872.727	2.99019
$45$	0	0
$46$	−599.039	−1.92007
$47$	273.742	0.849560	0.424780	−	0.905297i	$-0.360351\pi$
$47$	273.742	0.849560	0.424780	+	0.905297i	$0.360351\pi$
$48$	0	0
$49$	0	0
$50$	1070.25	3.02712
$51$	0	0
$52$	179.237	0.477994
$53$	255.936	0.663312	0.331656	−	0.943400i	$-0.392393\pi$
$53$	255.936	0.663312	0.331656	+	0.943400i	$0.392393\pi$
$54$	0	0
$55$	−1111.94	−2.72608
$56$	0	0
$57$	0	0
$58$	1088.20	2.46359
$59$	−168.833	−0.372545	−0.186272	−	0.982498i	$-0.559641\pi$
$59$	−168.833	−0.372545	−0.186272	+	0.982498i	$0.559641\pi$
$60$	0	0
$61$	195.141	0.409595	0.204797	−	0.978804i	$-0.434346\pi$
$61$	195.141	0.409595	0.204797	+	0.978804i	$0.434346\pi$
$62$	902.749	1.84918
$63$	0	0
$64$	−712.200	−1.39102
$65$	−228.366	−0.435774
$66$	0	0
$67$	515.148	0.939334	0.469667	−	0.882844i	$-0.344374\pi$
$67$	515.148	0.939334	0.469667	+	0.882844i	$0.344374\pi$
$68$	−169.113	−0.301587
$69$	0	0
$70$	0	0
$71$	319.048	0.533295	0.266648	−	0.963794i	$-0.414084\pi$
$71$	319.048	0.533295	0.266648	+	0.963794i	$0.414084\pi$
$72$	0	0
$73$	−635.518	−1.01893	−0.509464	−	0.860492i	$-0.670156\pi$
$73$	−635.518	−1.01893	−0.509464	+	0.860492i	$0.670156\pi$
$74$	157.845	0.247961
$75$	0	0
$76$	−731.048	−1.10338
$77$	0	0
$78$	0	0
$79$	−852.002	−1.21339	−0.606694	−	0.794935i	$-0.707505\pi$
$79$	−852.002	−1.21339	−0.606694	+	0.794935i	$0.707505\pi$
$80$	−636.244	−0.889178
$81$	0	0
$82$	2496.08	3.36153
$83$	264.419	0.349684	0.174842	−	0.984597i	$-0.444059\pi$
$83$	264.419	0.349684	0.174842	+	0.984597i	$0.444059\pi$
$84$	0	0
$85$	215.467	0.274949
$86$	1116.01	1.39933
$87$	0	0
$88$	1890.88	2.29055
$89$	914.197	1.08882	0.544409	−	0.838820i	$-0.316754\pi$
$89$	914.197	1.08882	0.544409	+	0.838820i	$0.316754\pi$
$90$	0	0
$91$	0	0
$92$	−1846.38	−2.09237
$93$	0	0
$94$	1303.60	1.43038
$95$	931.428	1.00592
$96$	0	0
$97$	455.582	0.476880	0.238440	−	0.971157i	$-0.423364\pi$
$97$	455.582	0.476880	0.238440	+	0.971157i	$0.423364\pi$
$98$	0	0
$99$	0	0
$100$	3298.76	3.29876
$101$	900.185	0.886849	0.443424	−	0.896312i	$-0.353763\pi$
$101$	900.185	0.886849	0.443424	+	0.896312i	$0.353763\pi$
$102$	0	0
$103$	114.380	0.109419	0.0547095	−	0.998502i	$-0.482577\pi$
$103$	114.380	0.109419	0.0547095	+	0.998502i	$0.482577\pi$
$104$	388.341	0.366153
$105$	0	0
$106$	1218.81	1.11680
$107$	−904.066	−0.816816	−0.408408	−	0.912799i	$-0.633916\pi$
$107$	−904.066	−0.816816	−0.408408	+	0.912799i	$0.633916\pi$
$108$	0	0
$109$	1446.61	1.27119	0.635596	−	0.772022i	$-0.280754\pi$
$109$	1446.61	1.27119	0.635596	+	0.772022i	$0.280754\pi$
$110$	−5295.23	−4.58982
$111$	0	0
$112$	0	0
$113$	10.2655	0.00854596	0.00427298	−	0.999991i	$-0.498640\pi$
$113$	10.2655	0.00854596	0.00427298	+	0.999991i	$0.498640\pi$
$114$	0	0
$115$	2352.47	1.90756
$116$	3354.10	2.68466
$117$	0	0
$118$	−804.006	−0.627244
$119$	0	0
$120$	0	0
$121$	2204.23	1.65607
$122$	929.292	0.689624
$123$	0	0
$124$	2782.49	2.01512
$125$	−1865.28	−1.33468
$126$	0	0
$127$	−1590.09	−1.11101	−0.555503	−	0.831514i	$-0.687474\pi$
$127$	−1590.09	−1.11101	−0.555503	+	0.831514i	$0.687474\pi$
$128$	−2652.39	−1.83157
$129$	0	0
$130$	−1087.51	−0.733701
$131$	−224.882	−0.149985	−0.0749924	−	0.997184i	$-0.523893\pi$
$131$	−224.882	−0.149985	−0.0749924	+	0.997184i	$0.523893\pi$
$132$	0	0
$133$	0	0
$134$	2453.21	1.58153
$135$	0	0
$136$	−366.406	−0.231022
$137$	2251.72	1.40421	0.702106	−	0.712072i	$-0.252243\pi$
$137$	2251.72	1.40421	0.702106	+	0.712072i	$0.252243\pi$
$138$	0	0
$139$	−3015.34	−1.83998	−0.919992	−	0.391937i	$-0.871805\pi$
$139$	−3015.34	−1.83998	−0.919992	+	0.391937i	$0.871805\pi$
$140$	0	0
$141$	0	0
$142$	1519.35	0.897896
$143$	726.051	0.424584
$144$	0	0
$145$	−4273.46	−2.44753
$146$	−3026.43	−1.71554
$147$	0	0
$148$	486.517	0.270212
$149$	−387.308	−0.212950	−0.106475	−	0.994315i	$-0.533956\pi$
$149$	−387.308	−0.212950	−0.106475	+	0.994315i	$0.533956\pi$
$150$	0	0
$151$	1643.03	0.885480	0.442740	−	0.896650i	$-0.354006\pi$
$151$	1643.03	0.885480	0.442740	+	0.896650i	$0.354006\pi$
$152$	−1583.91	−0.845212
$153$	0	0
$154$	0	0
$155$	−3545.17	−1.83713
$156$	0	0
$157$	693.668	0.352616	0.176308	−	0.984335i	$-0.443585\pi$
$157$	693.668	0.352616	0.176308	+	0.984335i	$0.443585\pi$
$158$	−4057.36	−2.04295
$159$	0	0
$160$	1728.03	0.853830
$161$	0	0
$162$	0	0
$163$	1745.54	0.838781	0.419390	−	0.907806i	$-0.362244\pi$
$163$	1745.54	0.838781	0.419390	+	0.907806i	$0.362244\pi$
$164$	7693.50	3.66318
$165$	0	0
$166$	1259.20	0.588754
$167$	3320.55	1.53863	0.769316	−	0.638868i	$-0.220597\pi$
$167$	3320.55	1.53863	0.769316	+	0.638868i	$0.220597\pi$
$168$	0	0
$169$	−2047.89	−0.932129
$170$	1026.09	0.462924
$171$	0	0
$172$	3439.80	1.52490
$173$	−54.1150	−0.0237820	−0.0118910	−	0.999929i	$-0.503785\pi$
$173$	−54.1150	−0.0237820	−0.0118910	+	0.999929i	$0.503785\pi$
$174$	0	0
$175$	0	0
$176$	2022.83	0.866345
$177$	0	0
$178$	4353.54	1.83321
$179$	−3720.54	−1.55355	−0.776776	−	0.629776i	$-0.783146\pi$
$179$	−3720.54	−1.55355	−0.776776	+	0.629776i	$0.783146\pi$
$180$	0	0
$181$	−1280.85	−0.525992	−0.262996	−	0.964797i	$-0.584711\pi$
$181$	−1280.85	−0.525992	−0.262996	+	0.964797i	$0.584711\pi$
$182$	0	0
$183$	0	0
$184$	−4000.43	−1.60280
$185$	−619.871	−0.246345
$186$	0	0
$187$	−685.041	−0.267889
$188$	4018.00	1.55874
$189$	0	0
$190$	4435.60	1.69364
$191$	940.010	0.356108	0.178054	−	0.984021i	$-0.443020\pi$
$191$	940.010	0.356108	0.178054	+	0.984021i	$0.443020\pi$
$192$	0	0
$193$	−1412.92	−0.526965	−0.263483	−	0.964664i	$-0.584871\pi$
$193$	−1412.92	−0.526965	−0.263483	+	0.964664i	$0.584871\pi$
$194$	2169.55	0.802910
$195$	0	0
$196$	0	0
$197$	3007.74	1.08778	0.543890	−	0.839156i	$-0.316951\pi$
$197$	3007.74	1.08778	0.543890	+	0.839156i	$0.316951\pi$
$198$	0	0
$199$	22.0447	0.00785280	0.00392640	−	0.999992i	$-0.498750\pi$
$199$	22.0447	0.00785280	0.00392640	+	0.999992i	$0.498750\pi$
$200$	7147.19	2.52691
$201$	0	0
$202$	4286.82	1.49316
$203$	0	0
$204$	0	0
$205$	−9802.29	−3.33962
$206$	544.693	0.184226
$207$	0	0
$208$	415.441	0.138489
$209$	−2961.32	−0.980090
$210$	0	0
$211$	−4881.84	−1.59279	−0.796397	−	0.604774i	$-0.793264\pi$
$211$	−4881.84	−1.59279	−0.796397	+	0.604774i	$0.793264\pi$
$212$	3756.65	1.21702
$213$	0	0
$214$	−4305.30	−1.37525
$215$	−4382.65	−1.39021
$216$	0	0
$217$	0	0
$218$	6888.96	2.14027
$219$	0	0
$220$	−16321.2	−5.00169
$221$	−140.691	−0.0428230
$222$	0	0
$223$	−144.053	−0.0432579	−0.0216290	−	0.999766i	$-0.506885\pi$
$223$	−144.053	−0.0432579	−0.0216290	+	0.999766i	$0.506885\pi$
$224$	0	0
$225$	0	0
$226$	48.8857	0.0143886
$227$	−4922.14	−1.43918	−0.719590	−	0.694400i	$-0.755670\pi$
$227$	−4922.14	−1.43918	−0.719590	+	0.694400i	$0.755670\pi$
$228$	0	0
$229$	15.7889	0.00455616	0.00227808	−	0.999997i	$-0.499275\pi$
$229$	15.7889	0.00455616	0.00227808	+	0.999997i	$0.499275\pi$
$230$	11202.8	3.21171
$231$	0	0
$232$	7267.10	2.05650
$233$	−727.505	−0.204551	−0.102276	−	0.994756i	$-0.532612\pi$
$233$	−727.505	−0.204551	−0.102276	+	0.994756i	$0.532612\pi$
$234$	0	0
$235$	−5119.34	−1.42106
$236$	−2478.14	−0.683530
$237$	0	0
$238$	0	0
$239$	63.6850	0.0172361	0.00861807	−	0.999963i	$-0.497257\pi$
$239$	63.6850	0.0172361	0.00861807	+	0.999963i	$0.497257\pi$
$240$	0	0
$241$	−3349.78	−0.895345	−0.447672	−	0.894198i	$-0.647747\pi$
$241$	−3349.78	−0.895345	−0.447672	+	0.894198i	$0.647747\pi$
$242$	10496.9	2.78829
$243$	0	0
$244$	2864.30	0.751508
$245$	0	0
$246$	0	0
$247$	−608.184	−0.156671
$248$	6028.62	1.54362
$249$	0	0
$250$	−8882.74	−2.24717
$251$	−4922.71	−1.23792	−0.618961	−	0.785422i	$-0.712446\pi$
$251$	−4922.71	−1.23792	−0.618961	+	0.785422i	$0.712446\pi$
$252$	0	0
$253$	−7479.30	−1.85857
$254$	−7572.26	−1.87057
$255$	0	0
$256$	−6933.49	−1.69275
$257$	2873.40	0.697422	0.348711	−	0.937230i	$-0.386619\pi$
$257$	2873.40	0.697422	0.348711	+	0.937230i	$0.386619\pi$
$258$	0	0
$259$	0	0
$260$	−3351.97	−0.799540
$261$	0	0
$262$	−1070.92	−0.252526
$263$	4367.70	1.02405	0.512023	−	0.858972i	$-0.328896\pi$
$263$	4367.70	1.02405	0.512023	+	0.858972i	$0.328896\pi$
$264$	0	0
$265$	−4786.35	−1.10952
$266$	0	0
$267$	0	0
$268$	7561.39	1.72345
$269$	−5354.90	−1.21373	−0.606866	−	0.794804i	$-0.707574\pi$
$269$	−5354.90	−1.21373	−0.606866	+	0.794804i	$0.707574\pi$
$270$	0	0
$271$	3042.25	0.681931	0.340966	−	0.940076i	$-0.389246\pi$
$271$	3042.25	0.681931	0.340966	+	0.940076i	$0.389246\pi$
$272$	−391.975	−0.0873786
$273$	0	0
$274$	10723.0	2.36424
$275$	13362.6	2.93016
$276$	0	0
$277$	−8580.03	−1.86110	−0.930549	−	0.366167i	$-0.880670\pi$
$277$	−8580.03	−1.86110	−0.930549	+	0.366167i	$0.880670\pi$
$278$	−14359.5	−3.09793
$279$	0	0
$280$	0	0
$281$	−953.058	−0.202330	−0.101165	−	0.994870i	$-0.532257\pi$
$281$	−953.058	−0.202330	−0.101165	+	0.994870i	$0.532257\pi$
$282$	0	0
$283$	7531.60	1.58200	0.791002	−	0.611813i	$-0.209560\pi$
$283$	7531.60	1.58200	0.791002	+	0.611813i	$0.209560\pi$
$284$	4683.00	0.978469
$285$	0	0
$286$	3457.57	0.714861
$287$	0	0
$288$	0	0
$289$	−4780.26	−0.972981
$290$	−20350.9	−4.12084
$291$	0	0
$292$	−9328.19	−1.86949
$293$	1325.09	0.264207	0.132104	−	0.991236i	$-0.457827\pi$
$293$	1325.09	0.264207	0.132104	+	0.991236i	$0.457827\pi$
$294$	0	0
$295$	3157.40	0.623155
$296$	1054.10	0.206988
$297$	0	0
$298$	−1844.42	−0.358538
$299$	−1536.07	−0.297100
$300$	0	0
$301$	0	0
$302$	7824.33	1.49086
$303$	0	0
$304$	−1694.45	−0.319681
$305$	−3649.41	−0.685129
$306$	0	0
$307$	8871.73	1.64930	0.824652	−	0.565640i	$-0.191371\pi$
$307$	8871.73	1.64930	0.824652	+	0.565640i	$0.191371\pi$
$308$	0	0
$309$	0	0
$310$	−16882.6	−3.09312
$311$	4531.09	0.826156	0.413078	−	0.910696i	$-0.364454\pi$
$311$	4531.09	0.826156	0.413078	+	0.910696i	$0.364454\pi$
$312$	0	0
$313$	−4316.61	−0.779518	−0.389759	−	0.920917i	$-0.627442\pi$
$313$	−4316.61	−0.779518	−0.389759	+	0.920917i	$0.627442\pi$
$314$	3303.35	0.593691
$315$	0	0
$316$	−12505.7	−2.22628
$317$	6697.55	1.18666	0.593331	−	0.804959i	$-0.297813\pi$
$317$	6697.55	1.18666	0.593331	+	0.804959i	$0.297813\pi$
$318$	0	0
$319$	13586.8	2.38468
$320$	13319.1	2.32675
$321$	0	0
$322$	0	0
$323$	573.831	0.0988508
$324$	0	0
$325$	2744.35	0.468397
$326$	8312.53	1.41223
$327$	0	0
$328$	16669.0	2.80607
$329$	0	0
$330$	0	0
$331$	−8065.66	−1.33936	−0.669681	−	0.742649i	$-0.733569\pi$
$331$	−8065.66	−1.33936	−0.669681	+	0.742649i	$0.733569\pi$
$332$	3881.16	0.641586
$333$	0	0
$334$	15812.9	2.59056
$335$	−9633.97	−1.57122
$336$	0	0
$337$	−480.216	−0.0776232	−0.0388116	−	0.999247i	$-0.512357\pi$
$337$	−480.216	−0.0776232	−0.0388116	+	0.999247i	$0.512357\pi$
$338$	−9752.34	−1.56940
$339$	0	0
$340$	3162.64	0.504465
$341$	11271.3	1.78995
$342$	0	0
$343$	0	0
$344$	7452.78	1.16810
$345$	0	0
$346$	−257.704	−0.0400412
$347$	−2237.12	−0.346094	−0.173047	−	0.984914i	$-0.555361\pi$
$347$	−2237.12	−0.346094	−0.173047	+	0.984914i	$0.555361\pi$
$348$	0	0
$349$	−2602.84	−0.399217	−0.199609	−	0.979876i	$-0.563967\pi$
$349$	−2602.84	−0.399217	−0.199609	+	0.979876i	$0.563967\pi$
$350$	0	0
$351$	0	0
$352$	−5493.99	−0.831905
$353$	6938.47	1.04617	0.523084	−	0.852281i	$-0.324781\pi$
$353$	6938.47	1.04617	0.523084	+	0.852281i	$0.324781\pi$
$354$	0	0
$355$	−5966.62	−0.892043
$356$	13418.7	1.99772
$357$	0	0
$358$	−17717.8	−2.61568
$359$	2153.11	0.316538	0.158269	−	0.987396i	$-0.449409\pi$
$359$	2153.11	0.316538	0.158269	+	0.987396i	$0.449409\pi$
$360$	0	0
$361$	−4378.42	−0.638347
$362$	−6099.59	−0.885600
$363$	0	0
$364$	0	0
$365$	11885.0	1.70436
$366$	0	0
$367$	−1848.98	−0.262986	−0.131493	−	0.991317i	$-0.541977\pi$
$367$	−1848.98	−0.262986	−0.131493	+	0.991317i	$0.541977\pi$
$368$	−4279.60	−0.606221
$369$	0	0
$370$	−2951.92	−0.414765
$371$	0	0
$372$	0	0
$373$	−44.2183	−0.00613816	−0.00306908	−	0.999995i	$-0.500977\pi$
$373$	−44.2183	−0.00613816	−0.00306908	+	0.999995i	$0.500977\pi$
$374$	−3262.27	−0.451037
$375$	0	0
$376$	8705.53	1.19403
$377$	2790.39	0.381200
$378$	0	0
$379$	3338.62	0.452489	0.226245	−	0.974071i	$-0.427355\pi$
$379$	3338.62	0.452489	0.226245	+	0.974071i	$0.427355\pi$
$380$	13671.6	1.84562
$381$	0	0
$382$	4476.47	0.599571
$383$	−8000.54	−1.06739	−0.533693	−	0.845678i	$-0.679196\pi$
$383$	−8000.54	−1.06739	−0.533693	+	0.845678i	$0.679196\pi$
$384$	0	0
$385$	0	0
$386$	−6728.54	−0.887238
$387$	0	0
$388$	6687.07	0.874960
$389$	−1125.83	−0.146740	−0.0733701	−	0.997305i	$-0.523375\pi$
$389$	−1125.83	−0.146740	−0.0733701	+	0.997305i	$0.523375\pi$
$390$	0	0
$391$	1449.30	0.187454
$392$	0	0
$393$	0	0
$394$	14323.3	1.83147
$395$	15933.6	2.02963
$396$	0	0
$397$	−182.743	−0.0231023	−0.0115512	−	0.999933i	$-0.503677\pi$
$397$	−182.743	−0.0231023	−0.0115512	+	0.999933i	$0.503677\pi$
$398$	104.980	0.0132216
$399$	0	0
$400$	7645.96	0.955745
$401$	−9397.81	−1.17033	−0.585167	−	0.810913i	$-0.698971\pi$
$401$	−9397.81	−1.17033	−0.585167	+	0.810913i	$0.698971\pi$
$402$	0	0
$403$	2314.85	0.286131
$404$	13213.0	1.62715
$405$	0	0
$406$	0	0
$407$	1970.78	0.240019
$408$	0	0
$409$	−2608.55	−0.315366	−0.157683	−	0.987490i	$-0.550402\pi$
$409$	−2608.55	−0.315366	−0.157683	+	0.987490i	$0.550402\pi$
$410$	−46680.0	−5.62283
$411$	0	0
$412$	1678.87	0.200758
$413$	0	0
$414$	0	0
$415$	−4944.99	−0.584916
$416$	−1128.33	−0.132983
$417$	0	0
$418$	−14102.3	−1.65015
$419$	9533.73	1.11158	0.555792	−	0.831322i	$-0.312415\pi$
$419$	9533.73	1.11158	0.555792	+	0.831322i	$0.312415\pi$
$420$	0	0
$421$	−11430.2	−1.32322	−0.661611	−	0.749848i	$-0.730127\pi$
$421$	−11430.2	−1.32322	−0.661611	+	0.749848i	$0.730127\pi$
$422$	−23248.1	−2.68175
$423$	0	0
$424$	8139.28	0.932260
$425$	−2589.34	−0.295532
$426$	0	0
$427$	0	0
$428$	−13269.9	−1.49866
$429$	0	0
$430$	−20870.8	−2.34065
$431$	6992.79	0.781510	0.390755	−	0.920495i	$-0.372214\pi$
$431$	6992.79	0.781510	0.390755	+	0.920495i	$0.372214\pi$
$432$	0	0
$433$	4730.76	0.525048	0.262524	−	0.964925i	$-0.415445\pi$
$433$	4730.76	0.525048	0.262524	+	0.964925i	$0.415445\pi$
$434$	0	0
$435$	0	0
$436$	21233.4	2.33233
$437$	6265.10	0.685814
$438$	0	0
$439$	8609.29	0.935988	0.467994	−	0.883732i	$-0.344977\pi$
$439$	8609.29	0.935988	0.467994	+	0.883732i	$0.344977\pi$
$440$	−35362.0	−3.83140
$441$	0	0
$442$	−669.991	−0.0721000
$443$	5271.91	0.565408	0.282704	−	0.959207i	$-0.408769\pi$
$443$	5271.91	0.565408	0.282704	+	0.959207i	$0.408769\pi$
$444$	0	0
$445$	−17096.7	−1.82126
$446$	−686.003	−0.0728323
$447$	0	0
$448$	0	0
$449$	−13513.8	−1.42039	−0.710197	−	0.704003i	$-0.751394\pi$
$449$	−13513.8	−1.42039	−0.710197	+	0.704003i	$0.751394\pi$
$450$	0	0
$451$	31164.8	3.25386
$452$	150.677	0.0156798
$453$	0	0
$454$	−23440.0	−2.42311
$455$	0	0
$456$	0	0
$457$	7288.39	0.746031	0.373016	−	0.927825i	$-0.378324\pi$
$457$	7288.39	0.746031	0.373016	+	0.927825i	$0.378324\pi$
$458$	75.1891	0.00767108
$459$	0	0
$460$	34529.8	3.49991
$461$	−5654.21	−0.571243	−0.285621	−	0.958343i	$-0.592200\pi$
$461$	−5654.21	−0.571243	−0.285621	+	0.958343i	$0.592200\pi$
$462$	0	0
$463$	4102.17	0.411758	0.205879	−	0.978577i	$-0.433995\pi$
$463$	4102.17	0.411758	0.205879	+	0.978577i	$0.433995\pi$
$464$	7774.24	0.777823
$465$	0	0
$466$	−3464.49	−0.344397
$467$	5104.11	0.505761	0.252880	−	0.967498i	$-0.418622\pi$
$467$	5104.11	0.505761	0.252880	+	0.967498i	$0.418622\pi$
$468$	0	0
$469$	0	0
$470$	−24379.1	−2.39260
$471$	0	0
$472$	−5369.22	−0.523598
$473$	13933.9	1.35451
$474$	0	0
$475$	−11193.3	−1.08123
$476$	0	0
$477$	0	0
$478$	303.277	0.0290200
$479$	4200.44	0.400674	0.200337	−	0.979727i	$-0.435796\pi$
$479$	4200.44	0.400674	0.200337	+	0.979727i	$0.435796\pi$
$480$	0	0
$481$	404.750	0.0383680
$482$	−15952.1	−1.50747
$483$	0	0
$484$	32353.9	3.03850
$485$	−8520.00	−0.797677
$486$	0	0
$487$	4300.99	0.400198	0.200099	−	0.979776i	$-0.435874\pi$
$487$	4300.99	0.400198	0.200099	+	0.979776i	$0.435874\pi$
$488$	6205.88	0.575670
$489$	0	0
$490$	0	0
$491$	1774.44	0.163095	0.0815474	−	0.996669i	$-0.474014\pi$
$491$	1774.44	0.163095	0.0815474	+	0.996669i	$0.474014\pi$
$492$	0	0
$493$	−2632.78	−0.240516
$494$	−2896.26	−0.263783
$495$	0	0
$496$	6449.33	0.583838
$497$	0	0
$498$	0	0
$499$	−6854.78	−0.614954	−0.307477	−	0.951556i	$-0.599485\pi$
$499$	−6854.78	−0.614954	−0.307477	+	0.951556i	$0.599485\pi$
$500$	−27378.7	−2.44883
$501$	0	0
$502$	−23442.7	−2.08426
$503$	6584.76	0.583698	0.291849	−	0.956464i	$-0.405730\pi$
$503$	6584.76	0.583698	0.291849	+	0.956464i	$0.405730\pi$
$504$	0	0
$505$	−16834.7	−1.48343
$506$	−35617.5	−3.12923
$507$	0	0
$508$	−23339.5	−2.03843
$509$	−4220.29	−0.367507	−0.183754	−	0.982972i	$-0.558825\pi$
$509$	−4220.29	−0.367507	−0.183754	+	0.982972i	$0.558825\pi$
$510$	0	0
$511$	0	0
$512$	−11799.2	−1.01847
$513$	0	0
$514$	13683.5	1.17423
$515$	−2139.05	−0.183025
$516$	0	0
$517$	16276.1	1.38457
$518$	0	0
$519$	0	0
$520$	−7262.49	−0.612464
$521$	5111.00	0.429783	0.214891	−	0.976638i	$-0.431060\pi$
$521$	5111.00	0.429783	0.214891	+	0.976638i	$0.431060\pi$
$522$	0	0
$523$	−14147.7	−1.18286	−0.591429	−	0.806357i	$-0.701436\pi$
$523$	−14147.7	−1.18286	−0.591429	+	0.806357i	$0.701436\pi$
$524$	−3300.83	−0.275186
$525$	0	0
$526$	20799.7	1.72416
$527$	−2184.09	−0.180532
$528$	0	0
$529$	3656.53	0.300528
$530$	−22793.3	−1.86807
$531$	0	0
$532$	0	0
$533$	6400.49	0.520142
$534$	0	0
$535$	16907.2	1.36629
$536$	16382.7	1.32020
$537$	0	0
$538$	−25500.8	−2.04353
$539$	0	0
$540$	0	0
$541$	21627.4	1.71874	0.859368	−	0.511357i	$-0.170857\pi$
$541$	21627.4	1.71874	0.859368	+	0.511357i	$0.170857\pi$
$542$	14487.6	1.14815
$543$	0	0
$544$	1064.60	0.0839050
$545$	−27053.5	−2.12632
$546$	0	0
$547$	−11453.9	−0.895305	−0.447652	−	0.894208i	$-0.647740\pi$
$547$	−11453.9	−0.895305	−0.447652	+	0.894208i	$0.647740\pi$
$548$	33050.9	2.57639
$549$	0	0
$550$	63634.6	4.93343
$551$	−11381.1	−0.879946
$552$	0	0
$553$	0	0
$554$	−40859.4	−3.13348
$555$	0	0
$556$	−44259.4	−3.37593
$557$	1041.42	0.0792219	0.0396109	−	0.999215i	$-0.487388\pi$
$557$	1041.42	0.0792219	0.0396109	+	0.999215i	$0.487388\pi$
$558$	0	0
$559$	2861.69	0.216523
$560$	0	0
$561$	0	0
$562$	−4538.61	−0.340658
$563$	19938.9	1.49259	0.746293	−	0.665618i	$-0.231832\pi$
$563$	19938.9	1.49259	0.746293	+	0.665618i	$0.231832\pi$
$564$	0	0
$565$	−191.978	−0.0142948
$566$	35866.6	2.66358
$567$	0	0
$568$	10146.4	0.749527
$569$	−14051.2	−1.03525	−0.517625	−	0.855608i	$-0.673184\pi$
$569$	−14051.2	−1.03525	−0.517625	+	0.855608i	$0.673184\pi$
$570$	0	0
$571$	−21502.4	−1.57591	−0.787956	−	0.615732i	$-0.788861\pi$
$571$	−21502.4	−1.57591	−0.787956	+	0.615732i	$0.788861\pi$
$572$	10657.0	0.779009
$573$	0	0
$574$	0	0
$575$	−28270.4	−2.05036
$576$	0	0
$577$	−16860.9	−1.21652	−0.608259	−	0.793739i	$-0.708132\pi$
$577$	−16860.9	−1.21652	−0.608259	+	0.793739i	$0.708132\pi$
$578$	−22764.3	−1.63818
$579$	0	0
$580$	−62726.2	−4.49063
$581$	0	0
$582$	0	0
$583$	15217.4	1.08103
$584$	−20210.8	−1.43207
$585$	0	0
$586$	6310.28	0.444839
$587$	3952.45	0.277913	0.138957	−	0.990298i	$-0.455625\pi$
$587$	3952.45	0.277913	0.138957	+	0.990298i	$0.455625\pi$
$588$	0	0
$589$	−9441.48	−0.660492
$590$	15036.0	1.04919
$591$	0	0
$592$	1127.66	0.0782883
$593$	15161.7	1.04995	0.524973	−	0.851119i	$-0.324076\pi$
$593$	15161.7	1.04995	0.524973	+	0.851119i	$0.324076\pi$
$594$	0	0
$595$	0	0
$596$	−5684.93	−0.390711
$597$	0	0
$598$	−7314.98	−0.500220
$599$	−7411.38	−0.505544	−0.252772	−	0.967526i	$-0.581342\pi$
$599$	−7411.38	−0.505544	−0.252772	+	0.967526i	$0.581342\pi$
$600$	0	0
$601$	−10776.0	−0.731388	−0.365694	−	0.930735i	$-0.619168\pi$
$601$	−10776.0	−0.731388	−0.365694	+	0.930735i	$0.619168\pi$
$602$	0	0
$603$	0	0
$604$	24116.5	1.62464
$605$	−41222.1	−2.77011
$606$	0	0
$607$	−11.6383	−0.000778225 0	−0.000389113	−	1.00000i	$-0.500124\pi$
$607$	−11.6383	−0.000778225 0	−0.000389113		1.00000i	$0.500124\pi$
$608$	4602.09	0.306973
$609$	0	0
$610$	−17379.0	−1.15353
$611$	3342.71	0.221329
$612$	0	0
$613$	15621.4	1.02927	0.514635	−	0.857410i	$-0.327928\pi$
$613$	15621.4	1.02927	0.514635	+	0.857410i	$0.327928\pi$
$614$	42248.5	2.77689
$615$	0	0
$616$	0	0
$617$	26376.5	1.72103	0.860517	−	0.509422i	$-0.170141\pi$
$617$	26376.5	1.72103	0.860517	+	0.509422i	$0.170141\pi$
$618$	0	0
$619$	21156.6	1.37375	0.686877	−	0.726774i	$-0.258981\pi$
$619$	21156.6	1.37375	0.686877	+	0.726774i	$0.258981\pi$
$620$	−52036.2	−3.37069
$621$	0	0
$622$	21577.7	1.39098
$623$	0	0
$624$	0	0
$625$	6790.67	0.434603
$626$	−20556.3	−1.31245
$627$	0	0
$628$	10181.7	0.646966
$629$	−381.888	−0.0242081
$630$	0	0
$631$	18757.1	1.18337	0.591686	−	0.806169i	$-0.298463\pi$
$631$	18757.1	1.18337	0.591686	+	0.806169i	$0.298463\pi$
$632$	−27095.4	−1.70537
$633$	0	0
$634$	31894.7	1.99795
$635$	29736.9	1.85838
$636$	0	0
$637$	0	0
$638$	64702.2	4.01502
$639$	0	0
$640$	49603.3	3.06366
$641$	3996.83	0.246280	0.123140	−	0.992389i	$-0.460704\pi$
$641$	3996.83	0.246280	0.123140	+	0.992389i	$0.460704\pi$
$642$	0	0
$643$	22237.8	1.36388	0.681940	−	0.731408i	$-0.261136\pi$
$643$	22237.8	1.36388	0.681940	+	0.731408i	$0.261136\pi$
$644$	0	0
$645$	0	0
$646$	2732.67	0.166433
$647$	−22992.2	−1.39709	−0.698545	−	0.715566i	$-0.746169\pi$
$647$	−22992.2	−1.39709	−0.698545	+	0.715566i	$0.746169\pi$
$648$	0	0
$649$	−10038.4	−0.607153
$650$	13069.0	0.788628
$651$	0	0
$652$	25621.2	1.53896
$653$	29699.3	1.77982	0.889912	−	0.456132i	$-0.150766\pi$
$653$	29699.3	1.77982	0.889912	+	0.456132i	$0.150766\pi$
$654$	0	0
$655$	4205.59	0.250880
$656$	17832.2	1.06133
$657$	0	0
$658$	0	0
$659$	388.786	0.0229817	0.0114909	−	0.999934i	$-0.496342\pi$
$659$	388.786	0.0229817	0.0114909	+	0.999934i	$0.496342\pi$
$660$	0	0
$661$	−9988.03	−0.587730	−0.293865	−	0.955847i	$-0.594942\pi$
$661$	−9988.03	−0.587730	−0.293865	+	0.955847i	$0.594942\pi$
$662$	−38409.9	−2.25505
$663$	0	0
$664$	8409.06	0.491468
$665$	0	0
$666$	0	0
$667$	−28744.7	−1.66867
$668$	48739.2	2.82302
$669$	0	0
$670$	−45878.4	−2.64543
$671$	11602.7	0.667536
$672$	0	0
$673$	19996.8	1.14535	0.572674	−	0.819783i	$-0.305906\pi$
$673$	19996.8	1.14535	0.572674	+	0.819783i	$0.305906\pi$
$674$	−2286.86	−0.130692
$675$	0	0
$676$	−30059.0	−1.71023
$677$	25123.8	1.42627	0.713135	−	0.701027i	$-0.247275\pi$
$677$	25123.8	1.42627	0.713135	+	0.701027i	$0.247275\pi$
$678$	0	0
$679$	0	0
$680$	6852.28	0.386431
$681$	0	0
$682$	53675.5	3.01370
$683$	23089.7	1.29356	0.646782	−	0.762675i	$-0.276114\pi$
$683$	23089.7	1.29356	0.646782	+	0.762675i	$0.276114\pi$
$684$	0	0
$685$	−42110.1	−2.34883
$686$	0	0
$687$	0	0
$688$	7972.88	0.441807
$689$	3125.29	0.172807
$690$	0	0
$691$	10755.9	0.592147	0.296074	−	0.955165i	$-0.404323\pi$
$691$	10755.9	0.592147	0.296074	+	0.955165i	$0.404323\pi$
$692$	−794.304	−0.0436343
$693$	0	0
$694$	−10653.5	−0.582710
$695$	56390.9	3.07774
$696$	0	0
$697$	−6038.96	−0.328181
$698$	−12395.1	−0.672152
$699$	0	0
$700$	0	0
$701$	31197.1	1.68088	0.840442	−	0.541901i	$-0.182295\pi$
$701$	31197.1	1.68088	0.840442	+	0.541901i	$0.182295\pi$
$702$	0	0
$703$	−1650.84	−0.0885670
$704$	−42345.9	−2.26700
$705$	0	0
$706$	33042.0	1.76141
$707$	0	0
$708$	0	0
$709$	28895.4	1.53059	0.765296	−	0.643678i	$-0.222592\pi$
$709$	28895.4	1.53059	0.765296	+	0.643678i	$0.222592\pi$
$710$	−28413.9	−1.50191
$711$	0	0
$712$	29073.3	1.53029
$713$	−23846.0	−1.25251
$714$	0	0
$715$	−13578.1	−0.710201
$716$	−54610.3	−2.85040
$717$	0	0
$718$	10253.5	0.532946
$719$	190.317	0.00987151	0.00493575	−	0.999988i	$-0.498429\pi$
$719$	190.317	0.00987151	0.00493575	+	0.999988i	$0.498429\pi$
$720$	0	0
$721$	0	0
$722$	−20850.7	−1.07477
$723$	0	0
$724$	−18800.4	−0.965069
$725$	51355.6	2.63076
$726$	0	0
$727$	5281.27	0.269424	0.134712	−	0.990885i	$-0.456989\pi$
$727$	5281.27	0.269424	0.134712	+	0.990885i	$0.456989\pi$
$728$	0	0
$729$	0	0
$730$	56598.4	2.86959
$731$	−2700.05	−0.136614
$732$	0	0
$733$	27817.2	1.40171	0.700853	−	0.713305i	$-0.252803\pi$
$733$	27817.2	1.40171	0.700853	+	0.713305i	$0.252803\pi$
$734$	−8805.12	−0.442783
$735$	0	0
$736$	11623.3	0.582122
$737$	30629.6	1.53088
$738$	0	0
$739$	−8553.43	−0.425768	−0.212884	−	0.977077i	$-0.568286\pi$
$739$	−8553.43	−0.425768	−0.212884	+	0.977077i	$0.568286\pi$
$740$	−9098.52	−0.451984
$741$	0	0
$742$	0	0
$743$	−14150.4	−0.698693	−0.349346	−	0.936994i	$-0.613596\pi$
$743$	−14150.4	−0.698693	−0.349346	+	0.936994i	$0.613596\pi$
$744$	0	0
$745$	7243.18	0.356201
$746$	−210.574	−0.0103347
$747$	0	0
$748$	−10055.1	−0.491511
$749$	0	0
$750$	0	0
$751$	−24243.3	−1.17796	−0.588981	−	0.808147i	$-0.700471\pi$
$751$	−24243.3	−1.17796	−0.588981	+	0.808147i	$0.700471\pi$
$752$	9313.05	0.451612
$753$	0	0
$754$	13288.3	0.641817
$755$	−30726.8	−1.48114
$756$	0	0
$757$	−13428.2	−0.644722	−0.322361	−	0.946617i	$-0.604476\pi$
$757$	−13428.2	−0.644722	−0.322361	+	0.946617i	$0.604476\pi$
$758$	15899.0	0.761844
$759$	0	0
$760$	29621.3	1.41379
$761$	−10525.9	−0.501398	−0.250699	−	0.968065i	$-0.580660\pi$
$761$	−10525.9	−0.501398	−0.250699	+	0.968065i	$0.580660\pi$
$762$	0	0
$763$	0	0
$764$	13797.5	0.653373
$765$	0	0
$766$	−38099.8	−1.79713
$767$	−2061.65	−0.0970558
$768$	0	0
$769$	−31855.3	−1.49380	−0.746900	−	0.664936i	$-0.768459\pi$
$769$	−31855.3	−1.49380	−0.746900	+	0.664936i	$0.768459\pi$
$770$	0	0
$771$	0	0
$772$	−20739.0	−0.966855
$773$	14812.7	0.689232	0.344616	−	0.938744i	$-0.388009\pi$
$773$	14812.7	0.689232	0.344616	+	0.938744i	$0.388009\pi$
$774$	0	0
$775$	42603.4	1.97466
$776$	14488.4	0.670237
$777$	0	0
$778$	−5361.38	−0.247063
$779$	−26105.5	−1.20067
$780$	0	0
$781$	18969.9	0.869136
$782$	6901.80	0.315611
$783$	0	0
$784$	0	0
$785$	−12972.5	−0.589821
$786$	0	0
$787$	−38359.3	−1.73744	−0.868718	−	0.495307i	$-0.835056\pi$
$787$	−38359.3	−1.73744	−0.868718	+	0.495307i	$0.835056\pi$
$788$	44147.9	1.99582
$789$	0	0
$790$	75878.1	3.41724
$791$	0	0
$792$	0	0
$793$	2382.91	0.106708
$794$	−870.251	−0.0388968
$795$	0	0
$796$	323.574	0.0144080
$797$	−26078.1	−1.15901	−0.579506	−	0.814968i	$-0.696754\pi$
$797$	−26078.1	−1.15901	−0.579506	+	0.814968i	$0.696754\pi$
$798$	0	0
$799$	−3153.90	−0.139646
$800$	−20766.3	−0.917751
$801$	0	0
$802$	−44753.8	−1.97046
$803$	−37786.5	−1.66060
$804$	0	0
$805$	0	0
$806$	11023.6	0.481751
$807$	0	0
$808$	28627.7	1.24643
$809$	−33877.7	−1.47228	−0.736142	−	0.676827i	$-0.763354\pi$
$809$	−33877.7	−1.47228	−0.736142	+	0.676827i	$0.763354\pi$
$810$	0	0
$811$	−15463.4	−0.669537	−0.334769	−	0.942300i	$-0.608658\pi$
$811$	−15463.4	−0.669537	−0.334769	+	0.942300i	$0.608658\pi$
$812$	0	0
$813$	0	0
$814$	9385.14	0.404114
$815$	−32644.0	−1.40303
$816$	0	0
$817$	−11671.9	−0.499813
$818$	−12422.3	−0.530973
$819$	0	0
$820$	−143879.	−6.12740
$821$	29146.6	1.23901	0.619503	−	0.784994i	$-0.287334\pi$
$821$	29146.6	1.23901	0.619503	+	0.784994i	$0.287334\pi$
$822$	0	0
$823$	−1831.30	−0.0775640	−0.0387820	−	0.999248i	$-0.512348\pi$
$823$	−1831.30	−0.0775640	−0.0387820	+	0.999248i	$0.512348\pi$
$824$	3637.50	0.153784
$825$	0	0
$826$	0	0
$827$	19801.5	0.832609	0.416304	−	0.909225i	$-0.363325\pi$
$827$	19801.5	0.832609	0.416304	+	0.909225i	$0.363325\pi$
$828$	0	0
$829$	−20707.6	−0.867557	−0.433778	−	0.901020i	$-0.642820\pi$
$829$	−20707.6	−0.867557	−0.433778	+	0.901020i	$0.642820\pi$
$830$	−23548.8	−0.984808
$831$	0	0
$832$	−8696.82	−0.362389
$833$	0	0
$834$	0	0
$835$	−62098.7	−2.57367
$836$	−43466.5	−1.79823
$837$	0	0
$838$	45401.1	1.87154
$839$	−28880.2	−1.18838	−0.594192	−	0.804323i	$-0.702528\pi$
$839$	−28880.2	−1.18838	−0.594192	+	0.804323i	$0.702528\pi$
$840$	0	0
$841$	27828.2	1.14101
$842$	−54432.5	−2.22787
$843$	0	0
$844$	−71656.0	−2.92240
$845$	38298.2	1.55917
$846$	0	0
$847$	0	0
$848$	8707.28	0.352605
$849$	0	0
$850$	−12330.8	−0.497580
$851$	−4169.46	−0.167952
$852$	0	0
$853$	−34006.5	−1.36502	−0.682509	−	0.730877i	$-0.739111\pi$
$853$	−34006.5	−1.36502	−0.682509	+	0.730877i	$0.739111\pi$
$854$	0	0
$855$	0	0
$856$	−28751.1	−1.14801
$857$	−27810.0	−1.10849	−0.554243	−	0.832355i	$-0.686992\pi$
$857$	−27810.0	−1.10849	−0.554243	+	0.832355i	$0.686992\pi$
$858$	0	0
$859$	−15538.4	−0.617186	−0.308593	−	0.951194i	$-0.599858\pi$
$859$	−15538.4	−0.617186	−0.308593	+	0.951194i	$0.599858\pi$
$860$	−64328.9	−2.55069
$861$	0	0
$862$	33300.7	1.31581
$863$	−34590.4	−1.36439	−0.682196	−	0.731169i	$-0.738975\pi$
$863$	−34590.4	−1.36439	−0.682196	+	0.731169i	$0.738975\pi$
$864$	0	0
$865$	1012.02	0.0397802
$866$	22528.6	0.884010
$867$	0	0
$868$	0	0
$869$	−50658.2	−1.97752
$870$	0	0
$871$	6290.58	0.244717
$872$	46005.0	1.78661
$873$	0	0
$874$	29835.4	1.15469
$875$	0	0
$876$	0	0
$877$	−13884.4	−0.534597	−0.267298	−	0.963614i	$-0.586131\pi$
$877$	−13884.4	−0.534597	−0.267298	+	0.963614i	$0.586131\pi$
$878$	40998.7	1.57590
$879$	0	0
$880$	−37829.7	−1.44914
$881$	50252.5	1.92173	0.960867	−	0.277009i	$-0.0893433\pi$
$881$	50252.5	1.92173	0.960867	+	0.277009i	$0.0893433\pi$
$882$	0	0
$883$	−22551.2	−0.859466	−0.429733	−	0.902956i	$-0.641392\pi$
$883$	−22551.2	−0.859466	−0.429733	+	0.902956i	$0.641392\pi$
$884$	−2065.07	−0.0785700
$885$	0	0
$886$	25105.6	0.951964
$887$	−28015.3	−1.06050	−0.530248	−	0.847842i	$-0.677901\pi$
$887$	−28015.3	−1.06050	−0.530248	+	0.847842i	$0.677901\pi$
$888$	0	0
$889$	0	0
$890$	−81417.1	−3.06641
$891$	0	0
$892$	−2114.43	−0.0793679
$893$	−13633.8	−0.510905
$894$	0	0
$895$	69579.0	2.59863
$896$	0	0
$897$	0	0
$898$	−64354.9	−2.39148
$899$	43318.2	1.60706
$900$	0	0
$901$	−2948.76	−0.109031
$902$	148411.	5.47844
$903$	0	0
$904$	326.462	0.0120110
$905$	23953.6	0.879827
$906$	0	0
$907$	26825.9	0.982071	0.491035	−	0.871140i	$-0.336619\pi$
$907$	26825.9	0.982071	0.491035	+	0.871140i	$0.336619\pi$
$908$	−72247.5	−2.64055
$909$	0	0
$910$	0	0
$911$	−9637.55	−0.350501	−0.175250	−	0.984524i	$-0.556074\pi$
$911$	−9637.55	−0.350501	−0.175250	+	0.984524i	$0.556074\pi$
$912$	0	0
$913$	15721.8	0.569896
$914$	34708.4	1.25607
$915$	0	0
$916$	231.751	0.00835945
$917$	0	0
$918$	0	0
$919$	3759.24	0.134936	0.0674678	−	0.997721i	$-0.478508\pi$
$919$	3759.24	0.134936	0.0674678	+	0.997721i	$0.478508\pi$
$920$	74813.3	2.68100
$921$	0	0
$922$	−26926.2	−0.961787
$923$	3895.95	0.138935
$924$	0	0
$925$	7449.20	0.264787
$926$	19535.2	0.693267
$927$	0	0
$928$	−21114.7	−0.746902
$929$	4162.56	0.147007	0.0735033	−	0.997295i	$-0.476582\pi$
$929$	4162.56	0.147007	0.0735033	+	0.997295i	$0.476582\pi$
$930$	0	0
$931$	0	0
$932$	−10678.4	−0.375302
$933$	0	0
$934$	24306.6	0.851536
$935$	12811.2	0.448097
$936$	0	0
$937$	39048.7	1.36144	0.680718	−	0.732545i	$-0.261668\pi$
$937$	39048.7	1.36144	0.680718	+	0.732545i	$0.261668\pi$
$938$	0	0
$939$	0	0
$940$	−75142.0	−2.60730
$941$	15015.4	0.520179	0.260089	−	0.965585i	$-0.416248\pi$
$941$	15015.4	0.520179	0.260089	+	0.965585i	$0.416248\pi$
$942$	0	0
$943$	−65933.6	−2.27687
$944$	−5743.91	−0.198038
$945$	0	0
$946$	66355.4	2.28055
$947$	12467.9	0.427825	0.213913	−	0.976853i	$-0.431379\pi$
$947$	12467.9	0.427825	0.213913	+	0.976853i	$0.431379\pi$
$948$	0	0
$949$	−7760.44	−0.265453
$950$	−53304.1	−1.82044
$951$	0	0
$952$	0	0
$953$	26584.1	0.903612	0.451806	−	0.892116i	$-0.350780\pi$
$953$	26584.1	0.903612	0.451806	+	0.892116i	$0.350780\pi$
$954$	0	0
$955$	−17579.4	−0.595662
$956$	934.773	0.0316242
$957$	0	0
$958$	20003.1	0.674605
$959$	0	0
$960$	0	0
$961$	6144.81	0.206264
$962$	1927.48	0.0645992
$963$	0	0
$964$	−49168.3	−1.64274
$965$	26423.5	0.881455
$966$	0	0
$967$	26613.6	0.885041	0.442521	−	0.896758i	$-0.354084\pi$
$967$	26613.6	0.885041	0.442521	+	0.896758i	$0.354084\pi$
$968$	70099.0	2.32755
$969$	0	0
$970$	−40573.5	−1.34303
$971$	10272.7	0.339512	0.169756	−	0.985486i	$-0.445702\pi$
$971$	10272.7	0.339512	0.169756	+	0.985486i	$0.445702\pi$
$972$	0	0
$973$	0	0
$974$	20481.9	0.673803
$975$	0	0
$976$	6638.96	0.217734
$977$	36897.7	1.20825	0.604127	−	0.796888i	$-0.293522\pi$
$977$	36897.7	1.20825	0.604127	+	0.796888i	$0.293522\pi$
$978$	0	0
$979$	54356.2	1.77450
$980$	0	0
$981$	0	0
$982$	8450.17	0.274599
$983$	21443.4	0.695766	0.347883	−	0.937538i	$-0.386901\pi$
$983$	21443.4	0.695766	0.347883	+	0.937538i	$0.386901\pi$
$984$	0	0
$985$	−56248.8	−1.81953
$986$	−12537.7	−0.404951
$987$	0	0
$988$	−8926.97	−0.287454
$989$	−29479.2	−0.947810
$990$	0	0
$991$	−16667.9	−0.534281	−0.267141	−	0.963658i	$-0.586079\pi$
$991$	−16667.9	−0.534281	−0.267141	+	0.963658i	$0.586079\pi$
$992$	−17516.3	−0.560628
$993$	0	0
$994$	0	0
$995$	−412.266	−0.0131354
$996$	0	0
$997$	−20248.7	−0.643213	−0.321606	−	0.946873i	$-0.604223\pi$
$997$	−20248.7	−0.643213	−0.321606	+	0.946873i	$0.604223\pi$
$998$	−32643.5	−1.03538
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bk.1.6		7
3.2	odd	2		1323.4.a.bh.1.2		7
7.3	odd	6		189.4.e.f.163.2	yes	14
7.5	odd	6		189.4.e.f.109.2	✓	14
7.6	odd	2		1323.4.a.bj.1.6		7
21.5	even	6		189.4.e.g.109.6	yes	14
21.17	even	6		189.4.e.g.163.6	yes	14
21.20	even	2		1323.4.a.bi.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.f.109.2	✓	14	7.5	odd	6
189.4.e.f.163.2	yes	14	7.3	odd	6
189.4.e.g.109.6	yes	14	21.5	even	6
189.4.e.g.163.6	yes	14	21.17	even	6
1323.4.a.bh.1.2		7	3.2	odd	2
1323.4.a.bi.1.2		7	21.20	even	2
1323.4.a.bj.1.6		7	7.6	odd	2
1323.4.a.bk.1.6		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q+4.76215 q^{2} +14.6781 q^{4} -18.7013 q^{5} +31.8020 q^{8} -89.0586 q^{10} +59.4578 q^{11} +12.2112 q^{13} +34.0213 q^{16} -11.5215 q^{17} -49.8054 q^{19} -274.500 q^{20} +283.147 q^{22} -125.792 q^{23} +224.740 q^{25} +58.1516 q^{26} +228.511 q^{29} +189.567 q^{31} -92.4014 q^{32} -54.8669 q^{34} +33.1458 q^{37} -237.181 q^{38} -594.740 q^{40} +524.149 q^{41} +234.349 q^{43} +872.727 q^{44} -599.039 q^{46} +273.742 q^{47} +1070.25 q^{50} +179.237 q^{52} +255.936 q^{53} -1111.94 q^{55} +1088.20 q^{58} -168.833 q^{59} +195.141 q^{61} +902.749 q^{62} -712.200 q^{64} -228.366 q^{65} +515.148 q^{67} -169.113 q^{68} +319.048 q^{71} -635.518 q^{73} +157.845 q^{74} -731.048 q^{76} -852.002 q^{79} -636.244 q^{80} +2496.08 q^{82} +264.419 q^{83} +215.467 q^{85} +1116.01 q^{86} +1890.88 q^{88} +914.197 q^{89} -1846.38 q^{92} +1303.60 q^{94} +931.428 q^{95} +455.582 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 31 q^{4} + q^{5} + 84 q^{8} - 12 q^{10} + 98 q^{11} + 124 q^{13} + 139 q^{16} - 30 q^{17} - 182 q^{19} + 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} + 245 q^{26} + 323 q^{29} - 26 q^{31}+ \cdots + 3730 q^{97}+O(q^{100}) \) 7 * q + q^2 + 31 * q^4 + q^5 + 84 * q^8 - 12 * q^10 + 98 * q^11 + 124 * q^13 + 139 * q^16 - 30 * q^17 - 182 * q^19 + 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 + 245 * q^26 + 323 * q^29 - 26 * q^31 + 398 * q^32 - 114 * q^34 - 112 * q^37 - 1015 * q^38 + 147 * q^40 + 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 - 288 * q^47 + 2576 * q^50 + 1075 * q^52 + 1353 * q^53 - 156 * q^55 - 81 * q^58 - 165 * q^59 - 56 * q^61 + 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 - 2625 * q^68 + 792 * q^71 - 1487 * q^73 + 2736 * q^74 - 1952 * q^76 - 1273 * q^79 + 2501 * q^80 + 2049 * q^82 + 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 - 1058 * q^89 + 3834 * q^92 - 1653 * q^94 + 3260 * q^95 + 3730 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more